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Paul Apisa

Publications and source records attributed to Paul Apisa.

At least 19 recordsLinked to original sources
arXiv2026

Components of strata of k-differentials and their orbit closures

We obtain a complete classification of components of strata of holomorphic and meromorphic k-differentials. We show that, when genus is at least two and outside of explicit exceptions when k < 4, there is one primitive nonhyperelliptic component unless k is odd and all singularities have even order, in which case there are two distinguished by their Arf invariant. The exceptions include new sporadic components of strata of cubic differentials. Our work provides a new proof of earlier results of Kontsevich-Zorich, Boissy, Lanneau, and Chen-Gendron when k = 1, 2. The proofs are almost purely algebraic, relying on the multiscale compactification of Bainbridge-Chen-Gendron-Grushevsky-Moller. This answers a question of Chen-Yu. We also show that for any component of a stratum of finite area $k$-differentials of positive genus, the smallest GL(2,R)-orbit closure containing all of its holonomy covers is as big as possible. This answers the positive genus analogue of a "long term goal" of Mirzakhani-Wright. The result also holds in genus zero strata that contain surfaces with Euclidean cylinders, thus addressing infinitely many cases of the original question of Mirzakhani-Wright.

math.AG↗
arXiv2025

Holonomy of affine surfaces

We identify the moduli space of complex affine surfaces with the moduli space of regular meromorphic connections on Riemann surfaces and show that it satisfies a corresponding universal property. As a consequence, we identify the tangent space of the moduli space of affine surfaces, at an affine surface X, with the first hypercohomology of a two-term sequences of sheaves on X. In terms of this identification, we calculate the derivative and coderivative of the holonomy map, sending an affine surface to its holonomy character. Using these formulas, we show that the holonomy map is a submersion at every affine surface that is not a finite-area translation surface, extending work of Veech. Finally, we introduce a holomorphic foliation of some strata of meromorphic affine surfaces, which we call the isoresidual foliation, along whose leave holonomy characters and certain residues are constant. We equip this foliation with a leafwise indefinite Hermitian metric, again extending work of Veech.

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arXiv2025

Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces

The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2,R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of Masur-Veech measure, i.e. an SL(2,R)-invariant Lebesgue class measure on strata or explicit covers thereof. This relies on a novel computation of cohomology with coefficients for the mapping class group. The computation produces a framed mapping class group invariant measure on representation varieties that naturally appear as the codomains of the periods maps that coordinatize strata.

math.GT↗
arXiv2024

Algebraically primitive invariant subvarieties with quadratic field of definition

We show that the only algebraically primitive invariant subvarieties of strata of translation surfaces with quadratic field of definition are the decagon, Weierstrass curves, and eigenform loci in genus two and the rank two example in the minimal stratum of genus four translation surfaces discovered by Eskin-McMullen-Mukamel-Wright.

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arXiv2023

Hurwitz-Hecke Invariant Subvarieties

We introduce a construction of affine invariant subvarieties in strata of translation surfaces whose input is purely combinatorial. We then show that this construction can be used to construct the Bouw-Moeller Teichmueller curves and the seven Eskin-McMullen-Mukamel-Wright rank two orbit closures. The construction is based on the theory of Hurwitz spaces and is inspired by work of Delecroix, Rueth, and Wright.

math.DS↗
arXiv2022

Moduli spaces of complex affine and dilation surfaces

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.

math.GT↗
arXiv2021

Billiards in right triangles and orbit closures in genus zero strata

The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials. Additionally, given a fixed set of angles, the orbit closure of the unfolding of all unit area rational parallelograms, isosceles trapezoids, and right trapezoids outside of a discrete set is determined.

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arXiv2021

Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy

We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani and Wright. As a corollary we show that outside of an explicit list of exceptions, if M is a GL(2,R)-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of M to absolute cohomology.

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arXiv2020

Generalizations of the Eierlegende-Wollmilchsau

We classify a natural collection of GL(2,R)-invariant subvarieties, which includes loci of double covers, the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces, and loci appearing naturally in the study of the complex geometry of Teichmuller space. This classification is the key input in subsequent work of the authors that classifies "high rank" invariant subvarieties, and in subsequent work of the first author that classifies certain invariant subvarieties with "Lyapunov spectrum as degenerate as possible". We also derive applications to the complex geometry of Teichmuller space and construct new examples, which negatively resolve two questions of Mirzakhani and Wright and illustrate previously unobserved phenomena for the finite blocking problem.

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arXiv2020

Reconstructing orbit closures from their boundaries

We introduce and study diamonds of GL(2,R)-invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneously considering two degenerations, and which provide a new tool for the classification of invariant subvarieties. We classify a surprisingly rich collection of diamonds where the two degenerations are contained in trivial invariant subvarieties. Our main results have been applied to classify large collections of invariant subvarieties; the statement of those results do not involve diamonds, but their proofs rely on them.

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arXiv2020

Periodic points on the regular and double $n$-gon surfaces

Using the transfer principle, we classify the periodic points on the regular $n$-gon and double $n$-gon translation surfaces and deduce consequences for the finite blocking problem on rational triangles that unfold to these surfaces.

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arXiv2017

Exceptional directions for the Teichm\"{u}ller geodesic flow and Hausdorff dimension

We prove that for every flat surface $\omega$, the Hausdorff dimension of the set of directions in which Teichm\"{u}ller geodesics starting from $\omega$ exhibit a definite amount of deviation from the correct limit in Birkhoff's and Oseledets' Theorems is strictly less than $1$. This theorem extends a result by Chaika and Eskin where they proved that such sets have measure $0$. We also prove that the Hausdorff dimension of the directions in which Teichm\"{u}ller geodesics diverge on average in a stratum is bounded above by $1/2$, strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly mixing IETs with permutation $(d, d-1, \dots, 1)$, where $d$ is an odd number, is exactly $1/2$ and strengthen a result by Avila and Leguil.

math.DS↗
arXiv2017

GL(2,R) Orbit Closures in Hyperelliptic Components of Strata

The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2, R) orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space.

math.DS↗
arXiv2017

GL(2,R)-Invariant Measures in Marked Strata: Generic Marked Points, Earle-Kra for Strata, and Illumination

We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmuller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2,R) orbit.

math.DS↗
arXiv2017

Periodic Points in Genus Two: Holomorphic Sections over Hilbert Modular Varieties, Teichmuller Dynamics, and Billiards

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varying families of points over orbifold covers of genus two Hilbert modular surfaces, solve the finite blocking problem on genus two translation surfaces, and show that there is at most one nonarithmetic rank two orbit closure in the minimal stratum in genus four.

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